GEOMETRIC PROBABILITY AND EXPECTED VALUE JIM RAHN WWW.JAMESRAHN.COM.

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Presentation transcript:

GEOMETRIC PROBABILITY AND EXPECTED VALUE JIM RAHN

A dart board consists of five concentric circles. The width of each of the rings is the same. If a dart is randomly launched at the dart board and it lands on the board. What is the probability the dart lands in each region? A Warm Up Activity

A dart board consists of five concentric circles. The width of each of the rings is the same. If a dart is randomly thrown at the dart board and it lands on the board. What is the probability the dart lands in each region? LABELArea of Disk Aπ(1) 2 = 1π Bπ(2) 2 - π(1) 2 = 3π Cπ(3) 2 - π(2) 2 = 5π Dπ(4) 2 - π(3) 2 = 7π Eπ(5) 2 - π(4) 2 = 9π Totalπ(5) 2 = 25π Theoretical Solution Using Geometry Prob. 1/25=0.04 3/25=0.12 5/25=0.20 7/25=0.25 9/25=0.36

Can we simulate tossing a dart at this dart board through a probability experiment? Place labeled disks (or cubes) with the indicated number of disks into a bag. Draw one disk at a time, record result, return disk to the bag. Repeat 100 times Experimental Results Label # of Disks in Bag A1 B3 C5 D7 E9 Total25 Simulating the Tossing of the Darts TagABCDETotal #

Possible results Experimental Results Simulating the Tossing of the Darts TagABCDETotal # Label # of Disks in Bag A1 B3 C5 D7 E9 Total25

Simulating the Toss with the graphing calculator Simulating the Tossing of the Darts- Method 2 TagABCDETotal # Select and classify 100 numbers Label # of Disks in Bag A1 B3 C5 D7 E9 Total25 Experimental Results Use random number generator on calculator randInt(1,25,5) A = 1 B = 2, 3, 4 C = 5, 6, 7, 8, 9 D = 10, 11, 12, 13, 14, 15, 16 E = 17, 18, 19, 20, 21, 22, 23, 24, 25

Possible results Simulating the Tossing of the Darts- Method 2 TagABCDETotal # Label # of Disks in Bag A1 B3 C5 D7 E9 Total25 Experimental Results

A dart board consists of five concentric circles. The width of each of the rings is the same. If a dart is randomly launched at the dart board and it lands on the board. What is the probability the dart lands in each region? FAIR GAME? Pay $1 to toss a dart. Hit A: Get back $3 Hit B: Get back $2 Hit C: Get back $1 Hit D or E: Lose $1 Expected Value Label # of Disks in Bag A1 B3 C5 D7 E9 Total25

You are at a state fair and are offered a “Game of Chance.” You pay $1 to toss a dart at this dartboard. Hit A: Get back $3, Hit B: Get back $2, Hit C: Get back $1 Hit D or E: Lose $1 Is this a “FAIR GAME?“ Expected Value Label Probability A1/25 B3/25 C5/25 D7/25 E9/25